GCSE Maths Practice: mutually-exclusive-events

Understanding Mutually Exclusive Events in Probability

In probability, events are described as mutually exclusive when they cannot happen at the same time. This means that if one event occurs, the other event definitely cannot. Understanding this idea is essential at GCSE level because it allows you to choose the correct rule when calculating probabilities.

For example, when rolling a standard six-sided die, getting a 2 and getting a 5 are mutually exclusive events. You cannot roll both numbers in a single roll. Because of this, the probability of getting a 2 or a 5 is found by adding the two individual probabilities together.

The Key Rule

When events are mutually exclusive, the probability of either event happening is calculated using the rule:

P(A or B) = P(A) + P(B)

This rule works because there is no overlap between the events. No outcomes are counted twice, so simple addition gives the correct result.

Worked Example 1

A spinner is divided into 8 equal sections numbered 1 to 8.

• The probability of landing on a 1 is 1/8.
• The probability of landing on a 6 is also 1/8.

Since the spinner cannot land on both numbers at once, these events are mutually exclusive. To find the probability of landing on a 1 or a 6, add the probabilities:

1/8 + 1/8 = 2/8

This fraction could then be simplified if needed.

Worked Example 2

A bag contains only red and blue counters.

• The probability of picking a red counter is 2/5.
• The probability of picking a blue counter is 3/5.

Picking a red counter and picking a blue counter cannot happen at the same time when only one counter is chosen. Therefore, these events are mutually exclusive. The probability of picking a red or blue counter is found by adding the two probabilities.

Common Mistakes to Avoid

• Adding probabilities when events are not mutually exclusive: If events can happen together, you must subtract the overlap instead of simply adding.
• Forgetting to check the question wording: Always look for phrases like “cannot happen together” or “only one outcome”.
• Mixing up ‘and’ with ‘or’: The word “or” usually suggests addition, but only when events are mutually exclusive.

Real-Life Applications

Mutually exclusive events appear often in everyday situations. For example, when choosing a meal option, you may select a vegetarian dish or a meat dish, but not both at the same time. In games, competitions, and surveys, understanding which outcomes exclude each other helps ensure probabilities are calculated correctly.

Frequently Asked Questions

How do I know if events are mutually exclusive?
Ask yourself whether both events could happen at the same time. If the answer is no, they are mutually exclusive.

Can probabilities of mutually exclusive events add up to more than 1?
No. Since probabilities represent chances, the total probability of all possible outcomes cannot exceed 1.

Is this topic tested often in GCSE Maths?
Yes. Mutually exclusive events are a core probability topic and often appear in foundation and higher exam questions.

Study Tip

Before using any probability rule, pause and decide whether the events can happen together. This single step will help you avoid most probability errors.